Materials Science and Engineering/Derivations/Models of Micro and Nanoscale Processing

First-Order Planar Growth Kinetics - The Linear Parabolic Model

File:Flux from gas phase to silicon surface.png

Oxide grows by indiffusion

Chemical Reaction

$Si+O_2\to Si{O}_2$

$Si+2H_{2}O\to Si{O}_2+2H_2$

Three Fluxes

Transport of the oxidant to the oxide surface

 $F_1=h_G(C_G-C_S)$

• $F_1$: flux in molecules
• $(C_G-C_S)$: concentration difference between gas flow and surface
• $h_G$: mass transfer coefficient

Equilibrium concentration of a gas species

The equilibrium concentration of a gas species dissolved in a solid is proportional to partial pressure of species at the surface.

$C_O=H{P}_S$

$C*=H{P}_G$

• $C*$:oxidant concentration in oxide that would be in equilibrium with $P_G$
• $P_G$: bulk gas pressure

From the ideal gas law:

$C_G=\frac{P_G}{kT}$

$C_S=\frac{P_S}{kT}$

$F_1=h(C*-C_O)$

Diffusion of oxidant through oxide to interface

In steady state,

 $F_2=-D\frac{\partial C}{\partial x}$

 $F_2=D\left(\frac{C_O-C_t}{x_O}\right)$

• $C_O$ and $C_I$: concetration at two interfaces
• $x_O$: oxide thickness

Oxygen and water seem to diffuse in different manners, though the effective diffusivities are of the same order.

Reaction at the Si/SiO2 interface

$F_3=k_S{C}_I$

• $k_S$: interface reaction rate constant

Equating three fluxes

With $F_1=F_2=F_3$

 $C_I=\frac{C*}{1+\frac{k_S}{h}+\frac{k_S{x}_O}{D}}$
 $C_I\approx \frac{C^{*}}{1+\frac{k_S{x}_O}{D}}$

 $C_O=\frac{C^{*}(1+\frac{k_S{x}_O}{D})}{1+\frac{k_S}{h}+\frac{k_S{x}_O}{D}}$
 $C_O\approx C^{*}$

The approximations are based on the observation that $h$ is very large. Gas absorption occurs rapidly compared with chemistry at interface.

Limiting cases

Reaction rate controlled - thin oxides

File:Limiting case in silicon oxidation - reaction rate controlled.png

Oxidant supplied to interface fast compared to that required to sustain the interface reaction

$C_I\approx C^{*}$

$k_S{x}_O/D<<1$

File:Limiting case in silicon oxidation - diffusion controlled regime.png

Diffusion controlled - thick oxides

$k_S{x}_O/D>>1$

$\frac{d{x}_O}{dt}=\frac{F}{N_1}$

$\frac{d{x}_O}{dt}=\frac{k_S{C}^{*}}{N_1(1+\frac{k_S}{h}+\frac{k_S{x}_O}{D})}$

• $N_1$: number of oxidant molecules incorporated

Integrate from initial oxide thickness $x_i$ to final thickness $x_o$:

$N_1\underset{x_i}{\overset{x_o}{\int }}1+\frac{k_S}{h}+\frac{k_S{x}_O}{D}d{x}_0=k_S{C}^{*}\underset{0}{\overset{t}{\int }}dt$

$\frac{x_O^2-x_i^2}{B}+\frac{x_O-x_i}{B/A}=t$

• $B=\frac{2D{C}^{*}}{N_1}$
• $\frac{B}{A}=\frac{C^{*}}{N_1(\frac{1}{k_S}+\frac{1}{h})}\frac{C^{*}{k}_S}{N_1}$

$\frac{x_O^2}{B}+\frac{x_O}{B/A}=t+\tau$

$\tau =\frac{x_i^2+A{x}_i}{B}$

$x_O=\frac{A}{2}(\sqrt{1+\frac{t+\tau }{A^2/4B}}-1)$

Limiting forms of the linear parabolic growth law

 $x_O\approx \frac{B}{A}(t+\tau )$

 $x_O^2\approx B(t+\tau )$